Volume 9, issue 1 (2009)

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Maps to the projective plane

Jerzy Dydak and Michael Levin

Algebraic & Geometric Topology 9 (2009) 549–568
Abstract

We prove the projective plane $ℝ{P}^{2}$ is an absolute extensor of a finite-dimensional metrizable space $X$ if and only if the cohomological dimension mod $2$ of $X$ does not exceed $1$. This solves one of the remaining difficult problems (posed by A N Dranishnikov) in Extension Theory. One of the main tools is the computation of the fundamental group of the function space $Map\left(ℝ{P}^{n},ℝ{P}^{n+1}\right)$ (based at the inclusion) as being isomorphic to either ${ℤ}_{4}$ or ${ℤ}_{2}\oplus {ℤ}_{2}$ for $n\ge 1$. Double surgery and the above fact yield the proof.

Keywords
absolute extensor, cohomological dimension, covering dimension, extension dimension, extension of maps, projective space
Mathematical Subject Classification 2000
Primary: 54F45
Secondary: 54C65, 55M10